In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.^[a][1]: 93

$${\displaystyle {\vec {F))=\mathbf {F} (\mathbf {r} )=\left\vert F(\mathbf {r} )\right\vert {\hat {\mathbf {r} ))}$$

${\textstyle {\vec {F))}$

${\textstyle {\hat {\mathbf {r} ))=\mathbf {r} /\|\mathbf {r} \|}$

Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant.^{[1]: 133–38}

Properties

Central forces that are conservative can always be expressed as the negative gradient of a potential energy:

$${\displaystyle \mathbf {F} (\mathbf {r} )=-\mathbf {\nabla } V(\mathbf {r} )\;{\text{, where ))V(\mathbf {r} )=\int _{|\mathbf {r} |}^{+\infty }F(r)\,\mathrm {d} r}$$

In a conservative field, the total mechanical energy (kinetic and potential) is conserved:

$${\displaystyle E={\tfrac {1}{2))m|\mathbf {\dot {r)) |^{2}+{\tfrac {1}{2))I|{\boldsymbol {\omega ))|^{2}+V(\mathbf {r} )={\text{constant))}$$

$${\displaystyle \mathbf {L} =\mathbf {r} \times m\mathbf {\dot {r)) ={\text{constant))}$$

It can also be shown that an object that moves under the influence of any central force obeys Kepler's second law. However, the first and third laws depend on the inverse-square nature of Newton's law of universal gravitation and do not hold in general for other central forces.

As a consequence of being conservative, these specific central force fields are irrotational, that is, its curl is zero, except at the origin:

$${\displaystyle \nabla \times \mathbf {F} (\mathbf {r} )=\mathbf {0} .}$$

Examples

Gravitational force and Coulomb force are two familiar examples with ${\displaystyle F(\mathbf {r} )}$ being proportional to 1/r² only. An object in such a force field with negative ${\displaystyle F(\mathbf {r} )}$ (corresponding to an attractive force) obeys Kepler's laws of planetary motion.

The force field of a spatial harmonic oscillator is central with ${\displaystyle F(\mathbf {r} )}$ proportional to r only and negative.

By Bertrand's theorem, these two, ${\displaystyle F(\mathbf {r} )=-k/r^{2))$and ${\displaystyle F(\mathbf {r} )=-kr}$, are the only possible central force fields where all bounded orbits are stable closed orbits. However, there exist other force fields, which have some closed orbits.